Graphics Reference
In-Depth Information
Each chapter also includes several homework problems. The goal of each problem
is to verify understanding of a basic concept, to understand and apply a formula,
or to fill in a derivation skipped in the main text. Most of these problems involve
simple linear algebra and calculus as a means to exercise these important muscles
in the service of a real computer vision scenario. Often, the derivations, or at least a
start on them, are found in one of the papers referenced in the chapter. On the other
hand, this topic doesn't have any problems like “implement algorithm X,” although
it should be easy for an instructor to specify programming assignments based on
the material in the main text. The emphasis here is on thoroughly understand-
ing the underlying mathematics, from which writing good code should (hopefully)
follow.
updated with links and commentary on new visual effects algorithms from academia
and industry, examples from behind the scenes of television and films, and demo
reels from visual effects artists and companies.
1.3
BACKGROUND AND PREREQUISITES
This topic assumes the reader has a basic understanding of linear algebra, such as
setting up a system of equations as a matrix-vector product and solving systems
of overdetermined equations using linear least-squares. These key concepts occur
repeatedly throughout the topic. Less frequently, we refer to the eigenvalues and
eigenvectors of a squarematrix, the singular value decomposition, andmatrix proper-
ties like positive definiteness. Strang's classic topic [
469
] is an excellent linear algebra
reference.
We also make extensive use of vector calculus, such as forming a Taylor series
and taking the partial derivatives of a function with respect to a vector of parameters
and setting them equal to zero to obtain an optimum. We occasionally mention
continuous partial differential equations, most of the time en route to a specific
discrete approximation. We also use basic concepts from probability and statistics
such as mean, covariance, and Bayes' rule.
Finally, the reader should have working knowledge of standard image process-
ing concepts such as viewing images as grids of pixels, computing image gradients,
creating filters for edge detection, and finding the boundary of a binary set of pixels.
On the other hand, this topic doesn't assume a lot of prior knowledge about com-
puter vision. In fact, visual effects applications form a great backdrop for learning
about computer vision for the first time. The topic introduces computer vision con-
cepts and algorithms naturally as needed. The appendixes include details on the
implementation of several algorithms common to many visual effects problems,
including dynamic programming, graph-cut optimization, belief propagation, and
numerical optimization. Most of the time, the sketches of the algorithms should
enable the reader to create a working prototype. However, not every nitty-gritty
implementation detail is provided, so many references are given to the original
research papers.
